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The 0-1 law for classical first-order logic states that, in a relational vocabulary, every formula is almost always true or almost always false on finite models. This theorem is due to Ronald Fagin and was proved in the 1970s building up on work started by Carnap. Given the failure of traditional model-theoretic properties such as compactness on finite models, it was quite remarkable to find a native of the finite setting using probabilistic techniques. In this talk, I will generalize the classical theorem to a many-valued context in the following form: for every formula there is a truth-value that the formula takes almost always or almost never on finite models. The new result will cover the cases of finitely-valued fuzzy logics such as Lukasiewicz, Gödel-Dummet and Product logics (and, of course, Boolean logic as an extreme case). This work also generalizes a theorem obtained in a more limited setting for the case of some Lukasiewicz logics by Robert Kosik and Chris Fermüller. (Joint work with Carles Noguera)
The finite tree property of intuitionistic logic entails completeness with respect to posets where each element, seen as a possibly partial situation, is under a maximal element, seen as a possible world containing the situation. This suggests a natural semantics for intuitionistic modal logic based on posets with a binary relation on the set of maximal elements. In this semantics, truth of modal formulas in a situation is determined by looking at worlds containing the situation and worlds accessible from them. In this paper we study modal logics arising from such a semantics. A general completeness-via-canonicty result is provided and various operations on such posets including filtrations are studied. Differences with respect to intuitionistic modal logics known from the literature are discussed. In the final part a completeness result for a version of intuitionistic propositional dynamic logic based on the framework is obtained and the logic is shown to be decidable.